Closed form singular value decomposition

ABSTRACT

The subject matter disclosed herein provides methods and apparatus for determining a singular value decomposition, providing feedback from a client station to a base station, and closed loop operation of a wireless system implementing multiple-input multiple-output (MIMO). The method may include determining one or more singular vectors using a closed form singular value decomposition. The one or more determined singular vectors may be provided to a precoder at the base station as feedback. The method may include aligning a phase of one or more singular vectors. The method may also include determining, at a client station, a plurality of singular vectors for channels used in a MIMO transmission from a base station to a client station. The client station may provide an indication to the base station regarding whether to use a singular value decomposition or a uniform channel decomposition. Related systems, apparatus, methods, and/or articles are also described.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of PCT Application No.PCT/US2009/049851 filed Jul. 7, 2009, which claims the benefit of U.S.Patent Application Ser. No. 61/078,766 filed Jul. 7, 2008. Thisapplication is also a continuation of PCT Application No.PCT/US2009/049852 filed Jul. 7, 2009, which claims the benefit of U.S.Patent Application Ser. No. 61/078,765 filed Jul. 7, 2008. Thisapplication is also a continuation of PCT Application No.PCT/US2009/049853 filed Jul. 7, 2009, which claims the benefit of U.S.Patent Application Ser. No. 61/078,767 filed Jul. 7, 2008. All of theabove referenced applications are hereby incorporated by referenceherein.

TECHNICAL FIELD

The subject matter described herein relates to wireless communications.

BACKGROUND

In signal processing associated with wireless devices, the singularvalue decomposition is frequently used to process signals. The singularvalue decomposition is a factorization of a rectangular real or acomplex matrix. For example, the singular value decomposition is adecomposition that may include determining a pseudoinverse, a leastsquares fitting of data, a matrix approximation, and determining therank, the range and/or the null space of a matrix. As such, the singularvalue decomposition is a computationally intensive operation, which inthe case of a wireless device, may be problematic. Moreover, in manyimplementations, the singular value decomposition is an iterativesolution, not of a closed form.

Multiple-input and multiple-output (MIMO) is typically used in wirelesscommunications to enhance performance, when compared to non-MIMOapproaches. For example, multiple antennas may be implemented at thetransmitter and/or the receiver to improve performance by providing, insome implementations, enhanced throughput and range. Often, theseperformance enhancements may be obtained without substantial increasesin transmitted power and/or bandwidth, hence the appeal of MIMO.However, MIMO typically comes at the cost of complex processing,including complex singular value decomposition processing, at thetransmitter and at the receiver

SUMMARY

The subject matter disclosed herein provides methods and apparatus fordetermining a singular value decomposition; providing feedback from aclient station to a base station; and closed loop operation of awireless system implementing multiple input multiple output (MIMO).

A method may include receiving a plurality of signals transmitted at abase station implementing a plurality of antennas configured for amultiple-input and multiple-output transmission. Moreover, one or moresingular vectors may be determined using a closed form singular valuedecomposition. The one or more determined singular vectors may beprovided to a precoder at the base station as feedback.

A method may include receiving, at a base station, a singular vectordetermined, at a client station, using a closed form singular valuedecomposition. The received singular vector may be used when precoding aplurality of signals for transmission using a multiple-input andmultiple-output transmission. Moreover, the precoded plurality ofsignals may be transmitted as the multiple-input and multiple-outputtransmission.

A method may align a phase of one or more singular vectors. The singularvectors may each represent one of a plurality of subcarriers of a band.Moreover, an average singular vector may be determined for the pluralityof subcarriers of the band, the aligned average singular vectorrepresentative of the plurality of subcarriers of the band. The alignedaverage singular vector may be provided, as feedback, to a precoder at abase station.

A method may include receiving, at a base station, an aligned averagesingular vector. Moreover, the received aligned average singular vectormay be used when precoding a plurality of signals for transmission usinga multiple-input and multiple-output transmission. Furthermore, theprecoded plurality of signals may be transmitted as the multiple-inputand multiple-output transmission.

A method may include determining, at a client station, a plurality ofsingular vectors for channels used in a multiple-input multiple-output(MIMO) transmission from a base station to a client station. Moreover,the client station may provide a first indication to the base station touse a singular value decomposition, when one of the singular vectors issubstantially larger (e.g., stronger) than the other singular vector.Furthermore, the client station may provide a second indication to thebase station to use a uniform channel decomposition, when one of thesingular vectors is not substantially larger than the other singularvector.

A method may include receiving, at a base station from a client station,a first indication to use a singular value decomposition, if one of thesingular vectors is substantially larger than the other singular vector.Moreover, the base station may receive from the client station, a secondindication to use a uniform channel decomposition, when one of thesingular vectors is not substantially larger than the other singularvector. Furthermore, the base station may be configured for transmissionbased on at least one of the first indication or the second indicationreceived from the client station.

The details of one or more variations of the subject matter describedherein are set forth in the accompanying drawings and the descriptionbelow. Features and advantages of the subject matter described hereinwill be apparent from the description and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

A more detailed understanding may be had from the following description,given by way of example in conjunction with the accompanying drawingswherein:

FIG. 1 depicts a block diagram of a network including client stationsand base stations;

FIG. 2 depicts a block diagram of a client station;

FIG. 3 depicts a process for determining a phase-aligned averagesingular vector for a band;

FIG. 4 depicts another process for determining a phase-aligned averagesingular vector;

FIG. 5 depicts a process for determining the closed form singular valuedecomposition;

FIG. 6 depicts a process for determining whether to use a singular valuedecomposition or a uniform channel decomposition based on the strengthof singular vectors;

FIG. 7 depicts a block diagram of a base station;

FIG. 8 depicts a process for using the phase-aligned average singularvector at a base station;

FIG. 9 depicts a process, at a base station, for using the singularvectors determined using closed form singular value decomposition;

FIG. 10 depicts a process for configuring a base station to use asingular value decomposition or a uniform channel decomposition; and

FIG. 11 depicts simulation results.

DETAILED DESCRIPTION

FIG. 1 is a simplified functional block diagram of an embodiment of awireless communication system 100. The wireless communication system 100includes a plurality of base stations 110A and 110B, each supporting acorresponding service or coverage area 112A and 112B. The base stationsare capable of communicating with wireless devices within their coverageareas. For example, the first base station 110A is capable of wirelesslycommunicating with a first client station 114A and a second clientstation 114B within the coverage area 112A. The first client station114A is also within the coverage area 112B and is capable ofcommunicating with the second base station 110B. In this description,the communication path from the base station to the client station isreferred to as a downlink 116A and the communication path from theclient station to the base station is referred to as an uplink 116B.

Although, for simplicity, only two base stations are shown in FIG. 1, atypical wireless communication system 100 may include a larger number ofbase stations. The base stations 110A and 110B can be configured ascellular base station transceiver subsystems, gateways, access points,radio frequency (RF) repeaters, frame repeaters, nodes, or any wirelessnetwork entry point.

The base stations 110A and 110B can be configured to support anomni-directional coverage area or a sectored coverage area. For example,the second base station 110B is depicted as supporting the sectoredcoverage area 112B. The coverage area 112B is depicted as having threesectors, 118A, 118B, and 118C. In typical embodiments, the second basestation 110B treats each sector 118 as effectively a distinct coveragearea.

Although only two client stations 114A and 114B are shown in thewireless communication system 100, typical systems are configured tosupport a large number of client stations. The client stations 114A and114B can be mobile, nomadic, or stationary units. The client stations114A and 114B are often referred to as, for example, mobile stations,mobile units, subscriber stations, wireless terminals, or the like. Aclient station can be, for example, a wireless handheld device, avehicle mounted device, a portable device, client premise equipment, afixed location device, a wireless plug-in accessory or the like. In somecases, a client station can take the form of a handheld computer,notebook computer, wireless telephone, personal digital assistant,wireless email device, personal media player, meter reading equipment orthe like and may include a display mechanism, microphone, speaker andmemory.

In a typical system, the base stations 110A and 110B also communicatewith each other and a network control module 124 over backhaul links122A and 122B. The backhaul links 122A and 122B may include wired andwireless communication links. The network control module 124 providesnetwork administration and coordination as well as other overhead,coupling, and supervisory functions for the wireless communicationsystem 100.

In some embodiments, the wireless communication system 100 can beconfigured to support both bidirectional communication andunidirectional communication. In a bidirectional network, the clientstation is capable of both receiving information from and providinginformation to the wireless communications network. Applicationsoperating over the bidirectional communications channel includetraditional voice and data applications. In a unidirectional network,the client station is capable of receiving information from the wirelesscommunications network, but may have limited or no ability to provideinformation to the network. Applications operating over theunidirectional communications channel include broadcast and multicastapplications. In one embodiment, the wireless system 100 supports bothbidirectional and unidirectional communications. In such an embodiment,the network control module 124 is also coupled to external entities via,for example, content link 126 (e.g., a source of digital video and/ormultimedia) and two-way traffic link 128.

The wireless communication system 100 can be configured to useOrthogonal Frequency Division Multiple Access (OFDMA) communicationtechniques. For example, the wireless communication system 100 can beconfigured to substantially comply with a standard system specification,such as IEEE 802.16 and its progeny or some other wireless standard suchas, for example, WiBro, WiFi, Long Term Evolution (LTE), or it may be aproprietary system. The subject matter described herein is not limitedto application to OFDMA systems or to the noted standards andspecifications. The description in the context of an OFDMA system isoffered for the purposes of providing a particular example only.

As used herein, IEEE 802.16 refers to one or more Institute ofElectrical and Electronic Engineers (IEEE) Standard for Local andmetropolitan area networks, Part 16: Air Interface for Fixed BroadbandWireless Access Systems, 1 Oct. 2004, IEEE Standard for Local andmetropolitan area networks, Part 16: Air Interface for Fixed and MobileBroadband Wireless Access Systems, 26 Feb. 2006, and any subsequentadditions or revisions to the IEEE 802.16 series of standards.

In some embodiments, downlinks 116A-B and uplink 116C each represent aradio frequency (RF) signal. The RF signal may include data, such asvoice, video, images, Internet Protocol (IP) packets, controlinformation, and any other type of information. When IEEE-802.16 isused, the RF signal may use OFDMA. OFDMA is a multi-user version oforthogonal frequency division multiplexing (OFDM). In OFDMA, multipleaccess is achieved by assigning to individual users groups ofsubcarriers (also referred to as subchannels or tones). The subcarriersare modulated using BPSK (binary phase shift keying), QPSK (quadraturephase shift keying), QAM (quadrature amplitude modulation), and carrysymbols (also referred to as OFDMA symbols) including data coded using aforward error-correction code.

In some embodiments, a base station is implemented using multiple-inputand multiple-output (MIMO). When MIMO is used, a base station mayinclude a plurality of antennas. For example, base station 110A may beconfigured for MIMO and include a precoder (described further below)coupled to two antennas for transmitting the MIMO transmission viadownlinks 116A-B. A client station may include a plurality of antennasto receive the MIMO transmission sent via downlinks 116A-B. The precoderis configured to perform “precoding,” which refers to beamforming tosupport MIMO transmission at each of the antennas (e.g., using singularvectors to weight orthogonal “eigen-beams” transmitted via each of theantennas). The client station may also combine the received signals,which may result in fewer errors and/or enhanced data transfer. Althoughthe examples given herein are made in the context of MIMO, other smartantenna techniques may be used as well including MISO (multiple input,single output) and SIMO (single input, multiple output).

Moreover, when MIMO is used, the base station may perform precoding(which may use additional information, such as, for example, channelestimation information) to code, for each antenna, one or more streamsof symbols for transmission over the corresponding antenna. In a closedloop feedback-based approach, the channel estimation information may beprovided by the client station to the base station. For example, aclient station may receive each of the downlinks 116A-B transmitted bythe antennas of the base station, decode the received downlink signals,determine channel estimation information for the decoded channels (e.g.,subcarriers) in each of the received downlink signals, and then provideto the base station the determined channel estimation information. Thechannel estimation information provided by the client station mayinclude singular vectors determined by the client station using asingular value decomposition (SVD). Although the channel estimationinformation is described as including singular vectors, the channelestimation information may also include other channel information, suchas the signal-to-noise ratio of a subcarrier, carrier-to-noise ratio, achannel covariance matrix, a channel matrix, and the like.

To determine the singular vectors, the client station may perform asingular value decomposition (SVD). The client station may provide, asfeedback, the singular vectors determined using the singular valuedecomposition. The singular vectors may be determined in a variety ofways. A channel estimator at the client station may determine thesingular vectors v₁ and v₂ (also referred to as u₁ and u₂), as describedbelow. Although the following describes determining singular vectors V₁and v₂ for the case of an N×2 matrix (i.e., when the client station has2 antennas), the closed form may be extended to client stations withother quantities of antennas. Further, a typical singular valuedecomposition is an iterative calculation, which may be computationallyintensive for a client station. As such, some or all of the subjectmatter described herein may relate to a closed form singular valuedecomposition, which may be less computationally intensive when comparedto typical, iterative singular value decomposition algorithms.

Alternatively or additionally, the singular vectors may be determinedfor each of the channels (e.g., subcarriers) used by the antennastransmitting from the base station to the client station. For example,the base station may include two antennas, each of which transmits overa channel comprising one or more subcarriers. The client station maythen determine singular vectors for the subcarriers. Moreover, thesubcarriers (as well as other channel information) may be used todetermine the so-called “strength” of the channel. For example, in thetwo antenna case, if the singular vector V₁ is substantially larger(e.g., when the eigen-subchannel of V_(i) is much larger thaneigen-subchannel of V₂) than the singular vector V₂, then the clientstation may provide an indication to the base station to use a singularvalue decomposition at the precoder. However, if the singular vector V₁is not substantially larger than the singular vector v₂, then the clientstation may provide an indication to the base station to use uniformchannel decomposition when precoding at the base station. In someimplementations, configuring the base station for a single stream ofsymbols using a singular value decomposition or a plurality of streamsof symbols using a uniform channel decomposition (e.g., based onfeedback from the client station indicating the “strength” of thesingular vectors) provides enhanced performance, compared to operationusing only a singular value decomposition or only a uniform channeldecomposition.

FIG. 2 depicts an exemplary client station, such as, for example, clientstation 114B. The client station 114B may include a plurality ofantennas 220A-B for receiving the downlinks 116A-B, each transmitted bya base station, such as base station 110A, which may implement MIMO asdescribed further below. Although the examples described herein refer totwo antennas at the base station and two antennas at the client station,other quantities of antennas may be used at the base station and theclient station. The client station 114B also includes a radio interface240, which may include other components, such as filters, converters(e.g., digital-to-analog converters and the like), symbol demappers, anInverse Fast Fourier Transform (IFFT) module, and the like, to processthe received MIMO transmission sent via downlinks 116A-B, to determinechannel estimation information, and to decode any data, such as thesymbols, carried by the downlinks. In some implementations, the clientstation 114B is also compatible with IEEE 802.16, OFDMA, and MIMOtransmissions (which may be sent via downlinks 116A-B). MIMOimplementations using other wireless technologies, such as LTE, WiBro,and the like, may also be implemented using the subject matter describedherein. The client station 114B may further include a channel estimator260 (described further below), a processor 220 for controlling clientstation 114B and for accessing and executing program code stored inmemory 225.

For each of the MIMO transmissions sent via downlinks 116A-B andreceived at each of antennas 220A-B, the channel estimator 260 maydetermine channel estimation information, such as singular vectorsdetermined using a singular value decomposition. For example, for eachof the subcarriers of a band, the channel estimator 260 may determinesingular vectors using a singular value decomposition. In someembodiments, rather than or in addition to performing a traditional,iterative singular value decomposition to determine the singularvectors, the channel estimator 260 may perform a closed form singularvalue decomposition. The closed form singular value decomposition may beused to determine singular vectors v₁ and v₂. For each of thesubcarriers, channel estimator 260 may determine one or more of thesingular vectors v₁ and v₂. Moreover, the channel estimator 260 may thenprovide, based on the strength of the determined singular vectors, anindication to the precoder at the base station to use a singular valuedecomposition or to use a uniform channel decomposition. When a singularvalue decomposition is used, the precoder at the base station provides,based on a singular value decomposition, a single stream of symbols forMIMO transmission. When uniform channel decomposition is used, theprecoder at the base station provides, based on a uniform channeldecomposition, a plurality of streams of symbols for MIMO transmission.The determined singular vectors (as well as any other channel estimationinformation) may then be provided as feedback by the client station to aprecoder at the base station (e.g., as a management message transmittedvia uplink 116C), which may be used at the precoder to configure for asingular value decomposition or a uniform channel decomposition.

In some embodiments, channel estimator 260 may determine the singularvector v_(i) based on the following:

ν₁ =h ₁*cos θ+h ₂*sin θe ^(jφ),

and may determine the singular vector v₂ based on the following:

ν₂ =h ₁*sin θ+h ₂*cos θe ^(jφ).

The determined singular vectors v₁ and v₂ may then be provided asfeedback to the base station (e.g., as a management message transmittedvia uplink 116C).

The following provides example implementations for determining thesingular vectors v₁ and v₂, although other approaches may be used aswell. Generally, analog feedback may provide advantages, such as lowcomplexity at the client station (e.g., a mobile station) and unboundedfeedback accuracy which may be particularly important for multi-userMIMO with uncorrelated antennas or multi-base station MIMO.

There are several options for feedback, such as using a full channelmatrix H, a covariance matrix H^(H)H , a strongest (e.g., largest) rightsingular vector, and two strongest right singular vectors. While feedingback the full channel matrix or channel covariance provides sufficientinformation for single user and/or multi-user MIMO, feeding back justthe first or two strongest eigenvectors may reduce feedback overhead.

For example, a 4 antenna base station requires 8 complex values forfeedback consisting of a full channel matrix or a covariance matrix, andonly 3 complex values for the strongest right singular vector option and5 complex values for the two strongest right singular vectors option. An8 antenna base station or two 4 antenna base stations in multi-basestation MIMO will require 16, 7, and 13 values for the full channelmatrix option, the strongest right singular vector option and the twostrongest right singular vectors option, respectively. In addition, atypical operation of closed loop MIMO in frequency division duplex mayrequire the client station (e.g., a mobile station) to send an initialestimate of the post precoding carrier-to-interference-plus-noise ratio(CINR).

To further illustrate, the following provides an example of how thesingular values of the channel may be calculated, as these willdetermine the post precoding signal-to-noise ratio (SNR).

Assume the channel matrix H is of dimension 2 by N (2×N), where N isgreater than or equal to 2 (N>=2). The singular values can be found fromthe eigenvalues of HH^(H), which is a 2×2 dimension matrix and which maybe solved based on the following two equations:

σ₁ ²+σ₂ ² =Tr(HH ^(H))

σ₁ ²σ₂ ² =det(HH ^(H)),

where Tr is trace and σi and σ₂ are the two singular values of H.

To calculate the singular value decomposition, the focus is on thematrix H^(H), which may be of size N×2. Then, the correct singularvectors that are of size 2 may be found and those singular vectors maybe used to find the left singular vectors that are the right singularvectors of H.

First, the form of H^(H) may be of the following form: H^(H)=VΣU^(H),where U may be generally expressed as follows:

$U = {\begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{\sin \; \theta \; ^{j\; \varphi}} & {{- \cos}\; \theta \; ^{j\; \varphi}}\end{pmatrix}.}$

By the definition of the singular value decomposition, the first columnof matrix U may be calculated according to the following equation:

$\theta,{\varphi = {\arg \; {\max\limits_{\theta,\varphi}{{H^{H}\begin{pmatrix}{\cos \; \theta} \\{\sin \; \theta \; ^{j\; \varphi}}\end{pmatrix}}}}}},$

where ∥x∥ represents a Euclidean norm. The maximum Euclidean norm is themaximal singular value σ₁.

Denoting the i^(th) row of matrix H as h_(i) and developing the aboveexpression, the following equation is formed:

${^{j\; \varphi} = \frac{h_{2}^{*}h_{1}}{{h_{2}^{*}h_{1}}}},$

which may be expressed (e.g., via substitution) as follows:

$\theta = {{\arg \; {\max\limits_{\theta}{{h_{1}}^{2}\cos^{2}\theta}}} + {{h_{2}}^{2}\sin^{2}\theta} + {2{{h_{2}^{*}h_{1}}}\sin \; {\theta cos}\; {\theta.}}}$

Differentiating and equating to zero, the following equation is formed:

${{\tan \; 2\; \theta} = \frac{2{{h_{2}^{*}h_{1}}}}{{h_{1}}^{2} - {h_{2}}^{2}}},$

after which a CORDIC rotation is used to calculate cos θ and sin θ.Using UH^(H)=VΣ, the strongest singular vector may be determined bynormalizing the following equation:

ν₁ =h ₁*cos θ+h ₂*sin θe ^(jφ).

At this point, the strongest right singular vector may be finalized. Tofinalize the two right strongest singular vectors, a solution to thefollowing equation may be found:

ν₂ =h ₁*sin θ+h ₂*cos θe ^(jφ), which may be normalized.

To illustrate further, the singular value decomposition (SVD) isgenerally done through iterative procedures. But for the special case ofan N×2 matrix, the decomposition may be done via a closed form solutionfor singular value decomposition. For example, given a matrix of thefollowing form:

H=[h₁

h₂]εC^(N×2), and then calculating the singular value decomposition asfollows:

H=UΣV*,

where U=[u₁

u₂]εC^(N×2) is a orthonormal matrix, and Σ is of the following form:

${\Sigma = \begin{pmatrix}\sigma_{1} & 0 \\0 & \sigma_{2}\end{pmatrix}},$

and V is of the following form:

$V = {\begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{\sin \; \theta \; ^{j\; \varphi}} & {{- \cos}\; {\theta }^{j\; \varphi}}\end{pmatrix}.}$

By the definition of the singular value decomposition, the first columnof V, which is parameterized by θ and φ, is calculated according to thefollowing equation:

$\theta,{\varphi = {\arg \; {\max\limits_{\theta,\; \varphi}{{H\begin{pmatrix}{\cos \; \theta} \\{\sin \; \theta \; ^{j\; \varphi}}\end{pmatrix}}}}}},$

where ∥x∥ represents the Euclidean norm and this maximum Euclidean normis the maximal singular value σ₁. Denoting a_(i), b_(i) as the ithelement of h₁ and h₂, respectively, the solution of the θ and φ equationabove may be derived based on the following equation:

$\begin{matrix}{\theta,{\varphi = {\arg \; {\max\limits_{\theta,\varphi}{\sum\limits_{i = 1}^{N}{{{a_{i}\cos \; \theta} + {b_{i}\sin \; \theta \; ^{j\; \varphi}}}}^{2}}}}}} \\{= {{\arg \; {\max\limits_{\theta,\; \varphi}{\sum\limits_{i = 1}^{N}{{a_{i}}^{2}\cos^{2}\theta}}}} + {\sum\limits_{i = 1}^{N}{{b_{i}}^{2}\sin^{2}\theta}} +}} \\{{2{Re}{\left\{ {\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}\sin \; {\theta cos}\; {\theta }^{{- j}\; \varphi}}} \right\}.}}}\end{matrix}$

Thus, the optimal φ=∠(Σ_(i=1) ^(N)a_(i)b_(i)*), which may be expressedas follows:

$^{j\; \varphi} = {\frac{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}}.}$

Further simplification of the θ and φ equation yields the following:

${\theta = {{\arg \; {\max\limits_{\theta}{\overset{N}{\sum\limits_{i = 1}}{{a_{i}}^{2}\cos^{2}\theta}}}} + {\sum\limits_{i = 1}^{N}{{b_{i}}^{2}\sin^{2}\; \theta}} + {2{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}}^{2}\sin \; \theta \; \cos \; \theta}}},$

and further constraining

$\theta \in \left\lbrack {0,\frac{\pi}{2}} \right)$

causes no loss of optimality, and equating the derivative to zero yieldsthe following:

${{\left( {{\sum\limits_{i = 1}^{N}{b_{i}}^{2}} - {\sum\limits_{i = 1}^{N}{a_{i}}^{2}}} \right)\sin \; 2\; \theta} + {2{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}}\cos \; 2\theta}} = 0.$

Hence, the following:

${\tan \; 2\theta} = {\frac{2{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}}}{{\sum\limits_{i = 1}^{N}{a_{i}}^{2}} - {\sum\limits_{i = 1}^{N}{b_{i}}^{2}}}.}$

The value of θ may be calculated based on the above equation, which mayhave only one solution in the interval 0 to π/2. However, since thesolution of interest is cos θ and sin θ rather than θ itself, thefollowing may be used to find a solution:

${\tan \; 2\; \theta} = {\frac{2\; \tan \; \theta}{1 - {\tan^{2}\theta}}.}$

When α is denoted as follows:

$\alpha = {\frac{{\sum\limits_{i = 1}^{N}{a_{i}}^{2}} - {\sum\limits_{i = 1}^{N}{b_{i}}^{2}}}{2{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}^{*}}}}}.}$

The following equation results:

${\frac{1 - {\tan^{2}\theta}}{2\; \tan \; \theta} = \alpha},$

which may lead to the unique solution in θ over the interval of 0 to π/2

$\left( {\theta \in \left\lbrack {0,\frac{\pi}{2}} \right)} \right),$

as follows:

tan θ=√{square root over (1+α²)}−α;

${{\cos \; \theta} = \frac{1}{\sqrt{1 + {\tan^{2}\theta}}}};$

and

sin θ=cos θ*tan θ.

Given the above cos θ and sin θ, V may be formed as follows:

$V = \begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} \\{\sin \; {\theta }^{j\; \varphi}} & {{- \cos}\; {\theta }^{j\; \varphi}}\end{pmatrix}$

and given the following:

UΣ=HV,

and also given the following:

u=h ₁ cos θ+h ₂ sin θe ^(jφ),

which may be the first column of UΣ, the largest singular value may havethe following form:

${\sigma_{1} = {{u} = \sqrt{{\sum\limits_{i = 1}^{N}{{a_{i}}^{2}\cos^{2}\theta}} + {\sum\limits_{i = 1}^{N}{{b_{i}}^{2}\sin^{2}\theta}} + {2{{\sum\limits_{i = 1}^{N}{a_{i}b_{i}}}}^{2}\cos \; \theta \; \sin \; \theta}}}},$

wherein ∥x∥ represents the Euclidean norm and the second equalityfollows from the θ symbol above. As such, the corresponding singularvector is:

$u_{1} = {\frac{u}{\sigma_{1}}.}$

Given the following equation,

ν=h ₁ sin θ−h ₂ cos θe ^(jφ),

which may be the second column of UΣ, and given σ₁ ²+σ₂ ²=∥H∥_(F) ²,where ∥x∥_(F) represents the Frobenius norm, the smaller singular valuemay be obtained as follows:

$\sigma_{2} = {\sqrt{{\sum\limits_{i = 1}^{N}{a_{i}}^{2}} + {\sum\limits_{i = 1}^{N}{b_{i}}^{2}} - \sigma_{1}^{2}}.}$

Thus, the corresponding singular vector may be:

$u_{2} = {\frac{v}{\sigma_{2}}.}$

Table 1 depicts example Matlab pseudo-code for determining the singularvectors as described above.

TABLE 1 Sample Pseudo-Code function  [U, S, V] = SVDNx2(H)${t_{1} = {\sum\limits_{i = 1}^{N}{{H\left( {i,1} \right)}}^{2}}};{\% \mspace{14mu} 2N\mspace{14mu} {real}\mspace{14mu} {multiplications}}$${t_{2} = {\sum\limits_{t = 1}^{N}{{H\left( {i,2} \right)}}^{2}}};{\% \mspace{14mu} 2N\mspace{14mu} {real}\mspace{14mu} {multiplications}}$${t_{3} = {\sum\limits_{i = 1}^{N}{{H\left( {i,1} \right)}*{{conj}\left( {H\left( {i,2} \right)} \right)}}}};{\% \mspace{14mu} 4N\mspace{14mu} {real}\mspace{14mu} {multiplications}}$t₄ = abs(t_(2 )); %  2  real  multip.  and  1  sqrte^(j φ) = t₃/t₄; %  1  divisiont₆ = (t₁ − t₂)/(2 * t₄); %  1  division${{\tan \mspace{14mu} \theta} = {\sqrt{1 + t_{0}^{2}} - t_{5}}};{\% \mspace{14mu} 1\mspace{14mu} {{multi}.\mspace{14mu} 1}\mspace{14mu} {sqrt}}$${{\cos \mspace{14mu} \theta} = {1/\sqrt{1 + {\tan^{2}\theta}}}};{{\% \mspace{14mu} 1\mspace{14mu} {{multi}.\mspace{14mu} 1}\mspace{14mu} {{sqrt}.\mspace{14mu} 1}\mspace{14mu} {{division}.\sin}\mspace{14mu} \theta} = {\tan \mspace{14mu} \theta*\cos \mspace{14mu} \theta}}\;;{\% \mspace{14mu} 1\mspace{14mu} {{multi}.}}$${t_{5} = {{t_{1}*\cos^{2}\theta} + {t_{2}*\sin^{2}\theta} + {2*t_{4}^{2}*\cos \mspace{14mu} \theta*\sin \mspace{14mu} \theta}}};\; {{\% \mspace{14mu} 7\mspace{14mu} {{multi}.\sigma_{1}}} = \sqrt{t_{6}}};{{\% \mspace{14mu} 1\mspace{14mu} {{sqrt}.\sigma_{2}}} = \sqrt{t_{1} + t_{2} - t_{5}}};{{\% \mspace{14mu} 1\mspace{14mu} {{sqrt}.S}} = \begin{pmatrix}\sigma_{1} & 0 \\0 & \sigma_{2}\end{pmatrix}};$ ${V = \begin{pmatrix}{\cos \mspace{14mu} \theta} & {\sin \mspace{14mu} \theta} \\{\sin \mspace{14mu} {\theta e}^{j\; \varphi}} & {{- \cos}\mspace{14mu} {\theta e}^{j\; \varphi}}\end{pmatrix}};$t₇ = h₂ * e^(j φ); %  4N  real  multi.u₁ = (h₁  cos   θ + t₇  sin   θ)/σ₁; %  4N  real  multi.  1  divisionu₂ = (h₁  sin   θ − t₇  cos   θ)/σ₁; %  4N  real  multi.  1  divisionU = [u₁, u₂].

Once the singular vectors (e.g., v₁ and v₂, which may also be referredto as singular vectors u₁ and u₂) are determined by channel estimator260, channel estimator 260 may provide the singular vectors to aprecoder at the base station. For example, channel estimator 260 mayprovide the determined singular vectors v₁ and v₂ to processor 220,which forwards the determined singular vectors v₁ and v₂ to the basestation in a management message via radio interface 240, one of theantennas 220A or 220B, and uplink 116C.

Rather than provide singular vectors for each of the subcarriers of aband (which may result in a considerable amount of the uplink 116C beingused for overhead), closed loop MIMO may instead provide, as feedback,one set of singular vectors v₁ and v₂ per band, wherein a typical bandis 400-800 KHz and spans 36-72 subcarriers. For example, that one set ofsingular vectors V₁ and v₂ per band may be determined for a centralsubcarrier of the band. However, this central subcarrier approach maynot accurately represent the band. The subject matter described hereinmay relate to determining, at a client station, a phase-aligned, averagesingular vector for the band. For example, to determine thephase-aligned, average singular vector, the client station aligns thephase of each of the singular vectors of the subcarriers of the band,and then determines the average of the singular vector over thesubcarriers of the band. The determined phase-aligned, average singularvector may be provided to the base station to configure a precoder for aMIMO transmission via downlinks 116A-B. In some implementations, thedetermined phase-aligned, average singular vector more accuratelyrepresents the band when compared to using the central subcarrier orusing an average across the band that is not phase aligned.

For each of the MIMO transmissions sent via downlinks 116A-B andreceived at each of antennas 220A-B, the channel estimator 260 maydetermine channel estimation information. For example, for each of thesubcarriers of a band, the channel estimator 260 may determine singularvectors using a singular value decomposition. In some embodiments,rather than perform a traditional, singular value decomposition todetermine the singular vectors for each of the subcarriers, the channelestimator 260 may perform a closed form singular value decomposition todetermine the singular vectors V₁ and v₂, as described above. For eachof the subcarriers, channel estimator 260 may determine one or more ofthe singular vectors v₁ and v₂.

To determine the phase-aligned, average singular vector, channelestimator 260 may select one of the subcarriers and then use thatsubcarrier's singular vector v₁ to phase align the other subcarrier'ssingular vectors v₁. Once phase-aligned, the channel estimator 260 maydetermine the average of the singular vectors v₁ across all of thesubcarriers of the band, and then normalize the average. In someembodiments, the channel estimator 260 may repeat the above processusing the determined phase-aligned average singular vector v₁ (e.g.,rather than selecting one of the subcarriers to align the phases of theother subcarriers, using the determined average and then aligning thephase of all of the other subcarriers relative to the determined averagesingular vector, and so forth). To determine the phase-aligned averagesingular vector for singular vector v₂, channel estimator 260 may repeatthe above-described process using the singular vector v₂ of each of thesubcarriers.

The following describes an exemplary implementation for determiningaverage singular vectors including phase-aligned average singular vectorfor singular vectors v₁ and v₂.

Specifically, rather than using the central subcarrier of the band, thefollowing four algorithms are described for designing a beamformingvector for a band which contains multiple subcarriers. The algorithmsdescribed below may be for one-stream schemes, but they may also beapplied to two-stream schemes.

In the first algorithm, let B be the set of subcarrier indices of theband. The first algorithm may include the following:

-   -   1. Compute the matrix

$\sum\limits_{i \in B}{H_{i}{H_{i}^{*}.}}$

-   -   2. Obtain the principle eigen-vector υ of

$\sum\limits_{i \in B}{H_{i}{H_{i}^{*}.}}$

The vector υ may be the beamforming vector to be used for the wholeband.

The second algorithm may include the following:

-   -   1. Compute the principle right singular vector υ_(i) of H_(i)        for iεB.    -   2. Obtain the matrix

$\sum\limits_{i \in B}{v_{i}{v_{i}^{*}.}}$

-   -   3. Calculate the principle eigen-vector υ of

${\sum\limits_{i \in B}{v_{i}v_{i}^{*}}},$

which may be the beamforming vector to be used for the whole band.

The third algorithm (which may include phase alignment) may include thefollowing:

-   -   1. Compute the principle right singular vector υ_(i) of H_(i)        for iεB.    -   2. Pick up a vector υ_(j) for an arbitrary jεB and align the        phase of other vectors by replacing

$\left. v_{j}\leftarrow{v_{i}\frac{v_{i}^{*}v_{j}}{{v_{i}^{*}v_{j}}}} \right.$

so that ν_(i)*ν_(j) is a positive number.

-   -   3. Calculate the norm-1 vector

$\frac{\sum\limits_{i \in B}v_{i}}{{\sum\limits_{i \in B}v_{i}}},$

which may be the beamforming vector to be used for the whole band.

The fourth algorithm (which may also include phase alignment) solves thefollowing optimization problem. Generally, the underlying process is tosolve the following minimization problem:

${\min\limits_{v,\varphi}{\sum{{{v_{i}^{{j\varphi}_{1}}} - v}}^{2}}},{{s.t.\mspace{14mu} {v}} = 1}$

wherein υ is the first or second singular vector of the channel in asubcarrier i. The fourth algorithm may work separately on each singularvector, each of which may not be limited in length. The optimalsolution, given known phase φ, may be as follows:

$v = {\sum\limits_{i \in B}{v_{i}{^{{j\varphi}_{1}}.}}}$

Hence, a solution may be found using the alternate minimization (AM)method as follows: (1) pick any subcarrier j and align the phases of allsingular vectors in the band relative to that subcarrier, i.e.,

$\left. v_{j}\leftarrow{v_{i}\frac{v_{i}^{*}v_{j}}{{v_{i}^{*}v_{j}}}} \right.;$

(2) calculate the average beamforming vector by normalizing the vector;and (3) repeat step 1 by using the vector calculated in step 2.Performing more iterations may generally lead to better performance, butin some cases one or two iterations of steps (1)-(3) may providesufficient accuracy.

FIG. 3 depicts a process 300 for determining the phase-aligned averagesingular vector. The description of process 300 will also refer to FIG.2. Channel estimator 260 at client station 114B may be configured todetermine the singular vectors for each of the plurality of subcarriersof a band (e.g., using a singular value decomposition). At 305, channelestimator 260 may align the phase of the singular vectors. At 310,channel estimator 260 may average the aligned singular vectors acrossthe band, thus yielding the phase-aligned average singular vector forthe band. In some embodiments, the channel estimator 260 normalizes thesingular vectors before determining the average. For example, thechannel estimator 260 may normalize the singular vector v₁ of each ofthe subcarriers of the band, and then may determine the average of thephase-aligned vectors v₁ of the subcarriers of the band. At 315, thephase-aligned average singular vector may be provided (e.g., sent) tothe base station, for example, to configure a precoder. In someimplementations, process 300 is stored as program code and may bestored, for example, at memory 225.

FIG. 4 depicts a process 400 for determining an aligned average singularvector, which is aligned in accordance with the phase of a referencesubcarrier. The description of process 400 will also refer to FIG. 2. At405, channel estimator 260 may compute, based on the channel estimate, av(i) (i.e., either a singular vector v₁ or a singular vector v₂) foreach of the i subcarriers of a multi-subcarrier band. At 410, channelestimator 260 selects one of the subcarriers as a so-called “referencesubcarrier.” Channel estimator 260 may align the phase (φ_(i)) of thesingular vectors of the subcarriers in accordance with the phase (φ_(i))of the singular vector of the reference subcarrier. At 415, channelestimator 260 may calculate the average aligned singular vector for theentire multi-subcarrier band from the sum of phase-aligned singularvectors v(i) of all the subcarriers (e.g., all i subcarriers). At 420,channel estimator 260 may repeat 410 and 415 using the average alignedsingular vector determined at 415 as the singular vector of thereference subcarrier. In some implementations, the phase alignment maybe performed as described above with respect to algorithms 3 and 4. Insome implementations, process 400 is stored as program code and may bestored, for example, at memory 225.

FIG. 5 depicts a process 500 for determining the closed form singularvalue decomposition. The description of process 500 will also refer toFIG. 2. At 505, a client station may receive a plurality of signals. Forexample, client station 114B may receive a plurality of RF signalstransmitted by base station 110B as a multiple-input and multiple-outputtransmission. At 510, a singular vector may be determined using a closedform singular value decomposition. For example, channel estimator 260may determine singular vectors v₁ and v₂ using the closed form singularvalue decomposition as described herein. At 515, the determined singularvector may be provided (e.g., sent) to the base station to configure aprecoder. In some implementations, process 500 is stored as program codeand may be stored, for example, at memory 225.

FIG. 6 depicts a process 600 for determining, at a client station,whether to use a singular value decomposition or a uniform channeldecomposition. The description of process 600 will also refer to FIG. 2.At 605, channel estimator 260 may determine one or more singular vectorsfor the channels (e.g., subcarriers). For example, if base station 110Btransmits using MIMO over two antennas, channel estimator 260 may use asingular value decomposition to determine the singular vectors (e.g.,singular vectors v₁ and v₂, also referred to as u₁ and u₂) for each ofthe channels.

At 610, channel estimator 260 determines whether one of the singularvectors is substantially larger (e.g., has a higher eigenvalue and isthus capable of more capacity) than the other singular vectors. Forexample, channel estimator 260 may normalize the determined singularvectors to have a range of zero to one. Given normalized vectors, asubstantially larger singular vector v₁ may have a value close to orequal to one, while another singular vector may have a value close tozero. Continuing with this example, at 615, the channel estimator 260may provide an indication to base station 110A to use a singular valuedecomposition at the precoder. If a singular value decomposition is usedat the base station, the precoder at base station 110B provides, basedon singular value decomposition, a single stream of symbols for MIMOtransmission. On the other hand, when the singular vector v₁ is notsubstantially larger than the other singular vector (e.g., singularvector v₁ does not have a value close to or equal to one), at 620, thechannel estimator 260 may provide an indication to the base station touse a uniform channel decomposition at the precoder. In someimplementations, process 600 is stored as program code and may bestored, for example, at memory 225.

FIG. 7 depicts a base station, such as base station 110A. The basestation 110A includes antennas 720A-B configured to transmit viadownlinks 116A-B and configured to receive uplink 116C via at least oneof antennas 720A-B. The base station 110A may further include a radiointerface 740 coupled to the antennas 720A-B, a precoder 760 (describedfurther below), a processor 730 for controlling base station 110A andfor accessing and executing program code stored in memory 735. The radiointerface 740 may further include other components, such as filters,converters (e.g., digital-to-analog converters and the like), mappers, aFast Fourier Transform (FFT) module, and the like, to generate symbolsfor a MIMO transmission via downlinks 116A-B and to receive the channelestimation information provided via uplink 116C; to receive thephase-aligned average singular vectors v₁ and v₂ for the band ofsubcarriers via uplink 116C; and/or to receive from a client station anindication of whether to use a singular value decomposition or a uniformchannel decomposition at the base station when transmitting using MIMOto the client station. The indication may serve as a form of rankadaptation In some implementations, the base station 110A may also becompatible with IEEE 802.16 and the RF signals of the MIMO downlinks116A-B and uplink 116C are configured in accordance with OFDMA. Theprecoder 760 may use the determined values of phase-aligned averagesingular vectors v₁ and v₂ as well as any other channel estimateinformation (including so-called “side information”) about thesubcarriers to precode each of the streams to be transmitted as a MIMOtransmission by antennas 720A-B.

The radio interface 740 may decode the uplink 116C carrying the singularvectors v₁ and v₂ and then provides the singular vectors v₁ and v₂ tothe precoder 760. Alternatively or additionally, the radio interface 740may decode the uplink 116C carrying the phase-aligned average singularvectors v₁ and v₂ and then provides the phase-aligned average singularvectors v₁ and v₂ to the precoder 760. The precoder 760 may use thesingular vectors v₁ and v₂ and/or the phase-aligned average singularvectors v₁ and v₂ (as well as any other channel estimation informationand/or side information provided as feedback by the client station tothe base station) to precode symbols for MIMO transmission via eachantenna 720A-B and downlinks 116A-B. The term “precoding” refers tobeamforming to support MIMO transmission at each of the antennas (e.g.,using singular vectors to weight orthogonal “eigen-beams” transmittedvia each of the antennas).

Alternatively or additionally, the radio interface 740 may decode theuplink 116C carrying an indication (e.g., a management message receivedfrom a client station) representative of whether the precoder isconfigured to perform a singular value decomposition or a uniformchannel decomposition. The radio interface 740 may also decode uplink116C carrying any channel estimation information (e.g., singular vectorsdetermined at the client station), which are provided to the precoder760. The precoder 760 receives the indication and may configure for asingular value decomposition or a uniform channel decomposition. When asingular value decomposition is used, the precoder at the base stationmay provide a single stream for MIMO transmission via the antennas720A-B. When uniform channel decomposition is used, the precoder 760uses any singular vectors v₁ and v₂ determined at the client station (aswell as any other channel estimation information provided as feedback bythe client station to the base station) to provide, based on a uniformchannel decomposition, a plurality of symbols streams (e.g., two symbolstreams) for MIMO transmission via each antenna 720A-B.

FIG. 8 depicts a process 800, at the base station, for using thephase-aligned average singular vectors determined at the client station.The description of process 800 will refer to FIGS. 2 and 7 as well. At805, a base station, such as base station 110A, receives thephase-aligned average singular vector (e.g., for v₁ and/or v₂)determined at a client station, such as client station 114B. At 810, thereceived phase-aligned average singular vector may be used at aprecoder, such as precoder 760, when precoding signals for transmissionusing multiple-input and multiple-output transmission. At 815, the basestation may transmit, as the multiple-input and multiple-outputtransmission, the precoded plurality of signals (e.g., symbols carriedvia downlink 116A-B). In some implementations, process 800 is stored asprogram code and may be stored, for example, at memory 735.

FIG. 9 depicts a process 900, at the base station, for using thesingular vectors determined using the closed form singular valuedecomposition. The description of process 900 will refer to FIGS. 2 and7 as well. At 905, a base station, such as base station 110A, receivesone or more singular vectors determined at a client station, such asclient station 114B, using the closed form singular value decompositiondescribed herein. At 910, the received singular vectors may be used at aprecoder, such as precoder 760, when precoding signals for transmissionusing multiple-input and multiple-output transmission. At 915, the basestation may transmit, as the multiple-input and multiple-outputtransmission, the precoded plurality of signals (e.g., symbols carriedvia downlink 116A-B). In some implementations, process 900 is stored asprogram code and may be stored, for example, at memory 735.

Moreover, in some implementations, the client station may feed back thelargest singular vector of the collaborating base station antennas. Thisis referred to as joint antenna processing multi-basestation MIMO in802.16m or coordinated multi-point transmission (COMP) in LTE. Forexample, if each base station has 4 antennas and 3 base stations arecollaborating, the singular vector has L=12 complex numbers. Thecomputation of the singular vector(s) for a client station with 2antennas may be performed as noted above. For example, the largestsingular vector for any subcarrier may be given by ν=h₁*cos θ+h₂*sinθe^(jφ), where h_(i) i=1,2 is the ith row of H, and the 2×L globaldownlink channel between the client station and all of the collaboratingbase stations is as follows:

$^{j\varphi} = \frac{h_{2}^{*}h_{1}}{{h_{2}^{*}h_{1}}}$

and

${\tan \; 2\theta} = {\frac{2{{h_{2}^{*}h_{1}}}}{{h_{1}}^{2} - {h_{2}}^{2}}.}$

The phase-aligned singular vectors of several subcarriers may beaveraged within a band to get the average singular vector. The resultingsingular vector is

${v = {\sum\limits_{k \in S}{v_{k}\frac{v_{k}^{*}v_{j}}{{v_{k}^{*}v_{j}}}}}},$

where j denotes one of the subcarriers.

FIG. 10 depicts a process 1000 for configuring a base station to use asingular value decomposition adaptation or a uniform channeldecomposition based on the strength singular vectors determined by aclient station. The description of process 1000 will refer to FIGS. 2and 7 as well. At 1005, a base station, such as base station 110A,receives an indication from a client station representative of whetherto use a singular value decomposition or a uniform channeldecomposition, when transmitting to the client station. For example,client station 114B may provide the indication as a management messagevia uplink 116C. At 1010, the received indication may be provided to aprecoder, such as precoder 760. If the indication corresponds to asingular value decomposition, the precoder 760 provides, based on asingular value decomposition, a single stream of symbols for MIMOtransmission via antennas 720A-B. When the indication corresponds touniform channel decomposition, the precoder 760 may use any singularvectors v₁ and v₂ determined at the client station (as well as any otherchannel estimation information provided as feedback by the clientstation to the base station) to provide, based on a uniform channeldecomposition, a plurality of symbols streams (e.g., two symbol streams)for MIMO transmission via antennas 720A-B. At 1015, the base station maytransmit to the client station based on the indication provided by theclient station. For example, the base station 110A may configuretransmission to the client station 114B based on whether the clientstation 114B has indicated that the base station 110A should transmit inaccordance with a singular value decomposition or a uniform channeldecomposition. In the case of a uniform channel decomposition, theclient station 114B may also provide channel estimation information,such as singular vectors, a channel covariance matrix, and the like. Insome implementations, process 1000 is stored as program code and may bestored, for example, at memory 735.

Moreover, typically, rank-2 operation decomposes the channel usingsingular value decomposition and transmits on the two eigen-subchannels.This orthogonalization of the channel may cause reduced performance insome implementations, which do not use matched modulation and codingschemes to each singular vector. To address that issue, theeigen-subchannels may be rotated mathematically such that the twosubchannels become two layers with identical output SINR when decodedusing a successive interference cancellation (SIC) receiver. This is theunderlying essence of uniform channel decomposition.

To further illustrate, rank adaptation may be performed using uniformchannel decomposition in a practical closed loop MIMO operation. Theclient station (e.g., mobile station) or base station may decide on thebest rank for the transmission according to the channel, SINR, and otherconsiderations using, for example, a capacity criterion. In the case ofrank-1 transmission, the strongest singular vector may be used. In thecase of rank-2 transmission, the precoder is calculated based on theuniform channel decomposition. In order to calculate the uniform channeldecomposition, information regarding the channel's right singularvectors and singular values are required. This is best facilitated infrequency division duplex by the client station feeding back analogfeedback of the channel or channel covariance matrix. Alternatively oradditionally, the two singular vectors and the ratio of the singularvalues may also be sent as feedback. Moreover, the feedback may befacilitated by using codebooks and feeding back for rank-2 one extravalue representing the ratio of the singular values.

FIG. 11 depicts plots of simulation results showing a comparison ofrank-2 transmission between the uniform channel decomposition method anda regular singular value decomposition in 2×2 and 2×4 configurationsusing a GSM TU channel, wherein one precoder per 9 subcarriers (bin) isused and 6 bits per second per hertz is the combined two stream spectralefficiency. FIG. 11 depicts the gain of uniform channel decompositionapproach.

The subject matter described herein may be embodied in systems,apparatus, methods, and/or articles depending on the desiredconfiguration. Base station 110A (or one or more components therein) canbe implemented using one or more of the following: a processor executingprogram code, an application-specific integrated circuit (ASIC), adigital signal processor (DSP), an embedded processor, a fieldprogrammable gate array (FPGA), and/or combinations thereof. Clientstation 114B (or one or more components therein) can be implementedusing one or more of the following: a processor executing program code,an application-specific integrated circuit (ASIC), a digital signalprocessor (DSP), an embedded processor, a field programmable gate array(FPGA), and/or combinations thereof. These various implementations mayinclude implementation in one or more computer programs that areexecutable and/or interpretable on a programmable system including atleast one programmable processor, which may be special or generalpurpose, coupled to receive data and instructions from, and to transmitdata and instructions to, a storage system, at least one input device,and at least one output device. These computer programs (also known asprograms, software, software applications, applications, components,program code, or code) include machine instructions for a programmableprocessor, and may be implemented in a high-level procedural and/orobject-oriented programming language, and/or in assembly/machinelanguage. As used herein, the term “machine-readable medium” refers toany computer program product, computer-readable medium, apparatus and/ordevice (e.g., magnetic discs, optical disks, memory, Programmable LogicDevices (PLDs)) used to provide machine instructions and/or data to aprogrammable processor, including a machine-readable medium thatreceives machine instructions as a machine-readable signal. Similarly,systems are also described herein that may include a processor and amemory coupled to the processor. The memory may include one or moreprograms that cause the processor to perform one or more of theoperations described herein.

Although a few variations have been described in detail above, othermodifications or additions are possible. In particular, further featuresand/or variations may be provided in addition to those set forth herein.For example, the above closed form singular value decomposition may beused in other applications, such as at the base station to determinesingular vectors for a sounder transmitted by the client station to thebase station. Moreover, the implementations described above may bedirected to various combinations and subcombinations of the disclosedfeatures and/or combinations and subcombinations of several furtherfeatures disclosed above. In addition, the logic flow depicted in theaccompanying figures and/or described herein does not require theparticular order shown, or sequential order, to achieve desirableresults. Other embodiments may be within the scope of the followingclaims.

1. A method comprising: receiving a plurality of signals transmitted bya base station implementing a plurality of antennas configured for amultiple-input and multiple-output transmission; determining, using aclosed form singular value decomposition, one or more singular vectors;and providing, as feedback, the one or more determined singular vectorsto a precoder at the base station.
 2. The method of claim 1, whereindetermining further comprises: determining, using the closed formsingular value decomposition, the one or more singular vectors, theclosed form singular value decomposition determined without iteration.3. The method of claim 1, wherein determining further comprises:determining, using the closed form singular value decomposition, a firstsingular vector based on the following equation:ν₁ =h ₁*cos θ+h ₂*sin θe ^(jφ); anda second singular vector based on the following equation:ν₂ =h ₁*sin θ+h ₂*cos θe ^(jφ).
 4. A method comprising: receiving, at abase station, a singular vector determined, at a client station, using aclosed form singular value decomposition; using the received singularvector when precoding a plurality of signals for transmission usingmultiple-input and multiple-output transmission; and transmitting, asthe multiple-input and multiple-output transmission, the precodedplurality of signals.
 5. A method comprising: aligning a phase of one ormore singular vectors, the singular vectors each representing one of aplurality of subcarriers of a band; determining, for the plurality ofsubcarriers of a band, an average singular vector for the alignedsingular vectors, the average singular vector representative of theplurality of subcarriers of the band; and providing, as feedback, thealigned average singular vector to a precoder at a base station.
 6. Themethod of claim 5, wherein aligning further comprises: selecting one ofthe subcarrriers as a reference subcarrier; and aligning a phase of eachof the singular vectors of the subcarrier of the band in accordance witha phase of a singular vector of the reference subcarrier, and whereindetermining the average singular vector further comprises: calculatingthe average aligned singular vector for all of the subcarriers of theband based on a sum of the phase-aligned singular vectors.
 7. A methodcomprising: receiving, at a base station, a phase-aligned averagesingular vector; using the received phase-aligned average singularvector when precoding a plurality of signals for transmission using amultiple-input and multiple-output transmission; and transmitting, asthe multiple-input and multiple-output transmission, the precodedplurality of signals.
 8. A method comprising: determining, at a clientstation, a plurality of singular vectors for channels used in amultiple-input multiple-output (MIMO) transmission from a base stationto the client station; providing, by the client station, a firstindication to the base station to use a singular value decomposition,when one of the singular vectors is substantially larger than the othersingular vector; and providing, by the client station, a secondindication to the base station to use a uniform channel decomposition,when one of the singular vectors is not substantially larger than theother singular vector.
 9. The method of claim 8, further comprising:receiving a single stream of symbols, when one of the singular vectorsis substantially larger than the other singular vector.
 10. The methodof claim 8, further comprising: receiving a plurality of streams ofsymbols, when one of the singular vectors is not substantially largerthan the other singular vector.
 11. A method comprising: receiving, at abase station from a client station, a first indication to use a singularvalue decomposition, when one of the singular vectors is substantiallylarger than the other singular vector; receiving, at the base stationfrom the client station, a second indication to use a uniform channeldecomposition, when one of the singular vectors is not substantiallylarger than the other singular vector; and configuring the base stationfor transmission based on at least one of the first indication or thesecond indication received from the client station.